Continuous Normalizing Flows

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1 paper in the last four weeks, against 2 the four weeks before. 0.0% of all new papers.

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Latest papers 19

Oct 7, 2026cs.LG

Seq-Flow: Efficient Probabilistic Forecasting with Self-Rollout Error Control

Many scientific forecasting tasks require updating a distribution over future trajectories as new observations arrive. Conventional diffusion and flow models generate each forecast from Gaussian noise, often at the cost of many sampling steps. Warm-start methods reuse earlier predictions to reduce this cost, but their models are not trained to perform the forecast update itself, which can compromise quality under few-step sampling. In this work, we introduce Seq-Flow, a conditional flow model whose ODE transports samples from the previous forecast distribution to the updated one. Because successive forecasts often differ only modestly, this transport starts from an informative distribution and can produce accurate updates with few flow evaluations. Recursive reuse also creates a challenge: errors in one forecast become errors in the initial states of subsequent flows. We address this with self-rollout training, in which a moving average copy of the model generates forecasts that initialize later training updates. Unlike self-forcing methods, which reuse generated outputs as conditioning context, Seq-Flow reuses them as the source of the next flow. Experiments On particle-accelerator beam spill forecasting show Seq-Flow reduces CRPS by 65% under a few-NFE sampling budget, while remaining competitive with strong baselines on fluid-dynamics forecasting tasks. Although trained on self-rollouts of at most four updates, Seq-Flow remains accurate over more than 400 consecutive updates. Our code is available at https://github.com/Graph-COM/Seq-Flow.

Yinan Huang, Shitij Govil, Bo Dai +1

Georgia Institute of Technology

Probabilistic Forecasting Probabilistic Time Series Forecasting Flow-Based Generative Modeling Continuous Normalizing Flows

Sep 24, 2026cs.SD

Off-manifold robustness in synthesizer inversion with joint distribution flow matching

Recent work on synthesizer inversion shows that generative models outperform deterministic approaches by explicitly modeling the ambiguity in mapping audio to parameters. Training such models, however, requires audio-parameter pairs, which are typically obtained by rendering sampled or preset parameters through the synthesizer itself. This creates a train-test mismatch that can degrade performance on off-manifold real-world recordings, for which ground-truth parameter annotations do not exist. To circumvent this obstacle, we propose to model the joint distribution of audio and parameters with a multi-modal continuous normalizing flow using independent noise schedules for each modality. This formulation allows us to train joint and conditional densities with paired synthesizer data, while unpaired real recordings can train the audio marginal alone, exposing the model to off-manifold signals without requiring parameter labels. Further, because the model learns to map from audio to parameters at all noise levels, we find that partially noising the audio reference at inference improves real-audio reconstruction, consistent with reducing sensitivity to distribution-specific detail while preserving coarse structure. Evaluating on Surge XT and Dexed, we find that modelling the joint distribution substantially improves both inversion of real-world and in-domain audio.

Ben Hayes

Sony Computer Science Laboratories Paris

Synthetic-to-Real Domain Adaptation Flow Matching Continuous Normalizing Flows

Sep 3, 2026math.NA

Learning Informative Prior with Infinite-Dimensional Continuous Normalizing Flow for Bayesian Inverse Problem

This paper addresses infinite-dimensional Bayesian inference for inverse problem of partial differential equations with model parameters in infinite-dimensional Hilbert space. To effectively incorporate prior information, we propose a novel continuous normalizing flows based infinite-dimensional model. Specifically, by introducing a well-defined neural ordinary differential equation in infinite-dimensional space, a simple reference measure can be transformed into a more complex measure which encodes the prior information. A corresponding theoretical framework is established to ensure the well-posedness of our proposed Bayesian prior in infinite-dimensional space. We also provide training methods of the prior for two distinct data settings, along with two sampling algorithms for the resulting Bayesian posterior. The proposed framework is applied to three representative inverse problems: the simple smooth inverse problem, inverse scattering problem, and the inverse heat conduction problem. Numerical experiments support the theoretical analysis and demonstrate the efficiency of the proposed algorithms.

Yang Zhao, Junxiong Jia, Tao Zhou

School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China · Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing, 100190, China

Normalizing Flows Bayesian Inverse Problems Posterior Sampling Continuous Normalizing Flows PDE Inverse Problems

Sep 1, 2026cs.CV

CQF-HMR: Continuous Quaternion Flows for Probabilistic 3D Human Mesh Recovery from a Single Image

Recovering 3D digital humans from a single 2D image is an ill-posed computer vision problem due to the loss of depth information. Probabilistic 3D human pose estimation compensates for this by estimating a set of 3D hypotheses from a prior distribution via generative models. However, most prior work focuses only on 3D keypoints, which often leads to implausible poses that are difficult to apply to downstream tasks, e.g. animation or digital humans. SMPL-based methods are more scalable thanks to the explicit body priors, but it requires more complex modeling of the generation process due to the non-additive nature of the joint rotations. In this work, we propose a novel approach for probabilistic 3D humans using quaternion-constrained continuous normalizing flows conditioned on 2D pose estimations. Our proposed quaternion flows show significant advantages over approaches using other rotation representations. Experiments demonstrate state-of-the-art results of our method on Human3.6M, particularly in ambiguous settings, and comparable pose estimation accuracy on challenging 3DPW and EMDB benchmarks.

Cuong Le, Bao-Long Tran, Pavlo Melnyk +3

Linköping University · Independent researcher.

Human Mesh Recovery 3D Human Pose Estimation Continuous Normalizing Flows

Aug 6, 2026cs.LG

Potential Matching Optimal Transport: Continuous Normalizing Flows for Exact $p$ -Wasserstein Dynamics

We introduce Potential Matching Optimal Transport (PMOT), a potential-flow framework for general

p

-cost optimal transport with

c_p(x,y)=\|x-y\|^p

. PMOT parameterizes the CNF velocity field with a scalar potential in the generalized Benamou--Brenier form for the chosen exponent

p

. It trains the potential gradient with a self-induced matching loss along straight bridges determined by the model's own endpoints, while allowing flexible terminal distribution matching. Our main result establishes zero-loss exactness: under the stated regularity, exact terminal matching, and uniqueness assumptions, any zero-loss solution satisfies the generalized Benamou--Brenier optimality system and recovers the corresponding

p

-optimal transport map and dynamics. On synthetic benchmarks, PMOT learns

p

-specific maps that agree with the corresponding

p

-matched OT references. It also remains competitive as a likelihood-based density model on high-dimensional tabular data, and an MMD-based color transformation experiment demonstrates flexible sample-based terminal matching.

Lishuo Zhang, Ruizhi Huang, Yang Yu +1

School of Mathematical Sciences, Shanghai Jiao Tong University · Institute of Natural Sciences, MOE-LSC, Shanghai Jiao Tong University

Flow Matching Continuous Normalizing Flows Optimal Transport

Jul 27, 2026cs.LG

Joint Flow Matching for Generator-Consistent Classification

We introduce Joint Flow Matching (JFM), a training framework for continuous normalising flows over multiple variables. Standard flow matching transports variables from noise to data simultaneously, offering no natural mechanism for forward and reverse conditional inference from a shared joint model. JFM resolves this by assigning opposite roles to each variable at the temporal endpoints. We prove that JFM produces a consistent joint distribution where that forward or reverse integration are conditionals of the same joint. We explore this consistency in the context of joint classification and generation as the basis for interpretability in discriminative-generative models. We validate JFM on conditional datasets producing competitive accuracy with inherently well-calibrated confidence scores without post-hoc calibration, and classifier-consistent image generation.

Hayden McAlister, Lech Szymanski

School of Computing University of Otago Dunedin, New Zealand

Flow Matching Continuous Normalizing Flows Flow-Based Generative Modeling Conditional Generative Modeling

Jul 8, 2026eess.SP

Stability of Flow Models for Graph Signals

Generating signals on graphs requires permutation-equivariant models that exhibit stability with respect to relative structural perturbations. While favorable stability properties of Graph Neural Networks (GNNs) have been well documented, it is unclear how structural errors propagate through the dynamics of continuous generative flow models that are gaining traction for graph signal generation. In this paper, we analyze continuous normalized flow models parameterized by GNNs and show that permutation equivariance is preserved for both the resulting continuous-time ordinary differential equations and their discrete numerical approximations used as graph signal samplers. Our primary contribution is to derive explicit stability bounds on the generated probability distributions, which quantify how relative graph perturbations affect the final sampled signals. Motivated by these theoretical bounds, we introduce a stability-promoting regularized flow matching strategy that actively penalizes the spatial Lipschitz constant of the vector field during model training. Experiments using synthetic smooth signals on stochastic block model graphs and real-world fMRI signals on brain connectomes demonstrate that this bound-oriented approach yields generative models that are more robust to structural noise, without sacrificing output quality.

Martin Schmidt, Gonzalo Mateos

Dept. of Electrical and Computer Engineering, University of Rochester, Rochester, NY 14627, USA

Graph Signal Processing Flow Matching Continuous Normalizing Flows Equivariant GNNs Flow-Based Generative Modeling

Jul 1, 2026cs.CL

Self-conditioned Flow Map Language Models via Fixed-point Flows

Self-conditioning is a core technique that enhances continuous flow-based language models, where the model learns to denoise generated text by conditioning on its own denoising estimate. While empirically successful, its performance improvements are poorly understood. Moreover, there is growing interest in the use of few-step generators based on flow maps, for which how to leverage self-conditioning is unclear. Here, we show that flow language models with self-conditioning perform a fixed-point iteration that improves generation through iterative refinement. We use this viewpoint to formulate fixed-point flows, a two-dimensional class of self-conditioned flows, where the first dimension represents the flow process and the second represents the fixed-point iteration. We show that fixed-point flows define valid flow maps, and show that they can be distilled from self-conditioned flow models by compressing both fixed-point iterations and the flow process, the former with fixed-point distillation and the latter with flow map distillation. Our resulting flow map language model, FMLM

^\star

, outperforms state-of-the-art self-conditioned models and few-step models in one- and few-step generation on OpenWebText. Code is available at https://github.com/Ugness/self-conditioned-fmlm.

Jaehoon Yoo, Wonjung Kim, Floor Eijkelboom +4

KAIST · University of Amsterdam · Carnegie Mellon University

Normalizing Flows One-Step Generative Modeling Generative Flow Networks Flow Map Learning Continuous Normalizing Flows Language Model Distillation+2

Jun 18, 2026cs.LG

Understanding Latent Flow Models for Tabular Data Synthesis: Targets, Paths, and Sampling

Synthetic tabular data enables microdata sharing in regulated domains, yet deploying continuous-time generative models requires balancing analytical utility, disclosure risk, and computational cost. Latent-space flow models are flexible, but theoretical equivalences across learning targets, probability paths, and sampling dynamics can translate into different behaviour under finite-step integration and explicit compute budgets. We present an empirical study of tabular latent flow models across seven datasets, evaluating velocity, score, noise, and posterior matching objectives under optimal transport (OT) and variance-preserving (VP) paths, ODE and SDE sampling, and varying integration budgets. Our contributions are threefold: (1) we show that the learning target largely determines the utility-risk operating regime, with velocity and posterior matching tending to yield higher utility, while score and noise matching tend to achieve lower disclosure risk; (2) we demonstrate that configuration and sampling choices shift performance, with midpoint often improving distributional fidelity and OT paths often tolerating earlier stopping than VP, enabling compute savings under fixed budgets or risk thresholds; and (3) we distil these findings into actionable defaults and practical configuration guidance to support pre-release model selection under disclosure risk and resource constraints. The code implementation and supplementary materials can be accessed in https://github.com/rulnasution/tabular-latent-flow/.

Bahrul Ilmi Nasution

Department of Social Statistics, The University of Manchester, UK

Synthetic Tabular Data Generation Continuous Normalizing Flows Flow-Based Generative Modeling Synthetic Data Evaluation

Jun 1, 2026cs.LG

Low-Pass Flow Matching

Flow Matching typically relies on white noise sources, a choice often misaligned with the power spectra of natural data, which tend to decay with frequency. To address this, we introduce Low-Pass Flow Matching, a variant of Flow Matching based on an operator-modulated interpolant. This formulation induces a time-varying spectral bias that transitions from the source spectrum to a frequency-decaying bias as the path approaches the data. We validate our method on unconditional image generation tasks, including the scientific Galaxy10 dataset. Empirically, we show that our method is particularly effective when paired with adaptive ODE solvers, where it improves or preserves sample quality while substantially reducing sampling cost compared to standard baselines.

Francesco M. Ruscio, T. Konstantin Rusch

ELLIS Institute Tübingen & · Max Planck Institute for Intelligent Systems & · AITHYRA +2

Flow Matching Image Generation Continuous Normalizing Flows Flow-Based Generative Modeling

May 30, 2026eess.AS

Local Diagnostics of Continuous Normalizing Flow for Out-of-Distribution Detection

We address the problem of out-of-distribution (OOD) detection for target observations embedded in a subspace of the high dimensional data space. Using continuous normalizing flows (CNFs), we propose a Lagrangian sub-flow (LSF) framework designed to isolate and estimate the density for the relevant components in the representation and using the remaining components as context. Through experimentation with models for speech synthesis, we show that CNFs, similarly to other deep generative models (DGMs), are susceptible to the "likelihood paradox", where high likelihood is erroneously assigned to OOD samples. This is attributed to the inductive bias of DGMs that prioritize low-level structural details over high-level semantic coherence. To mitigate this phenomenon, we propose a number of geometric diagnostic signals based on the velocity field over the sub-flow trajectory. Based on these signals, we design metrics for the challenging task of zero-shot phoneme-level mispronunciation detection. Finally, we demonstrate the superiority of these metrics compared to likelihood-based methods on a real-world mispronunciation detection benchmark.

Xinwei Cao, Mengxuan Lu, Torbjørn Svendsen +1

Department of Electronic Systems Norwegian University of Science and Technology Trondheim, Norway · School of Future Technology South China University of Technology Guangzhou, China

Speech Processing Distribution Shift Detection Continuous Normalizing Flows OOD Detection

May 27, 2026cs.LG

Parameter-Efficient Generative Modeling with Controlled Vector Fields

We introduce a continuous-time generative modeling framework, motivated by the Chow-Rashevskii theorem, that builds expressive flows from a small set of fixed vector fields and learned scalar controls. Instead of learning an unconstrained high-dimensional vector field, our framework constructs the velocity by modulating fixed vector fields with learned scalar control functions. When the fixed fields are bracket-generating, their Lie algebra spans the ambient space, providing a mechanism for expressive transport with only a small number of learned control channels and offering a parameter-efficient geometric alternative to standard vector-field parameterizations. This decoupled formulation yields a structured and interpretable generative model in which the number of learned scalar output channels can be chosen independently of the ambient dimension. We formulate an expressivity principle showing that, under suitable controllability and well-posedness assumptions, such controlled flows can transport a source distribution to a target distribution. We train the resulting model using a continuous-normalizing-flow likelihood objective and present proof-of-concept experiments on synthetic distributions.

Peyman Morteza

Generative Modeling Continuous Normalizing Flows Flow-Based Generative Modeling

May 25, 2026cs.LG

Two-Parameter Flows for Learning Population Dynamics of Physical Systems

This work addresses the problem of learning the dynamics of high-dimensional probability densities over time using unlabeled samples, without assuming access to trajectory information. We introduce two-parameter flows that learn only sampling-time transports from a base distribution to each marginal and then extract a physics-time velocity by regressing on coupled synthetic trajectories. We prove that the resulting physics-time dynamics are unique and inherit regularity from the sampling-time transports. Because we can build on standard, well-developed conditional flow matching techniques for learning the base-to-marginal transports, our approach scales to high dimensions and avoids per-step optimal-transport couplings, while allowing admissible non-gradient dynamics that can naturally explain rotational or circulating physics phenomena.

Paul Schwerdtner, Tobias Blickhan, Benjamin Peherstorfer

Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, USA.

Dynamical Systems Continuous Normalizing Flows Flow-Based Generative Modeling

May 15, 2026stat.ML

StAD: Stein Amortized Divergence for Fast Likelihoods with Diffusion and Flow

Diffusion and flow-based models are ubiquitously used for generative modelling and density estimation. They admit a deterministic probability flow ordinary differential equation (PF-ODE), analogous to continuous normalizing flows (CNFs), which describes the transport of the probability mass. Obtaining the likelihood from these models is of interest to many workflows, especially Bayesian analysis, and requires solving the trace of the Jacobian to compute the divergence of the learned PF-ODE, which is either

\mathcal{O}(D^2)

to compute exactly or

\mathcal{O}(D)

with a noisy estimate. We introduce StAD, a new distillation method to predict and learn the divergence of the PF-ODE using the Langevin-Stein operator without ever computing the Jacobian. We show that our method is competitive with the Hutchinson and Hutch++ on CIFAR-10, ImageNet and other density estimation tasks, consistently improving the variance and speed of the likelihood predictions compared to the Hutchinson. We additionally show our method will generalize to a varied class of generative models, and show that under some regularity conditions these learned vector fields can be made to satisfy the Stein class.

Gurjeet Jagwani, Stephen Thorp, Sinan Deger +1

Institute of Astronomy and Kavli Institute for Cosmology, University of Cambridge, Cambridge, UK · Research Computing Services, University of Cambridge, Cambridge, UK · Cavendish Laboratory, Department of Physics, University of Cambridge, Cambridge, UK +1

Continuous Normalizing Flows Flow-Based Generative Modeling Density Estimation Diffusion Models

May 11, 2026cs.LG

Language Modeling with Hyperspherical Flows

Discrete Diffusion Language Models progressed rapidly as an alternative to autoregressive (AR) models, motivated by their parallel generation abilities. However, for tractability, discrete diffusion models sample from a factorized distribution, which is less expressive than AR. Recent Flow Language Models (FLMs) apply continuous flows to language, transporting noise to data with a deterministic ODE that avoids factorized sampling. FLMs operate on one-hot vectors whose dimension scales with the vocabulary size, making FLMs costly to train. Moreover, since all distinct one-hot embeddings are equidistant in

\ell_2

, adding Gaussian noise does not have a clear semantic interpretation (unlike images, where Gaussian noise progressively degrades structure). We introduce

\mathbb{S}

-FLM, a latent FLM in the hypersphere.

\mathbb{S}

-FLM generates sequences by rotating vectors in

\mathbb{S}^{d-1}

along a velocity field learned with cross-entropy, avoiding the overhead of materializing one-hot vectors. Previous FLMs match AR in Generative Perplexity (Gen.\ PPL), but samples with high likelihood are not necessarily correct in verifiable domains such as math and code.

\mathbb{S}

-FLM substantially improves continuous flow language models on large-vocabulary reasoning and closes the gap to masked diffusion under standard-temperature sampling (

T=1

), while a gap remains under optimized low-temperature (

T=0.1

) decoding.

Justin Deschenaux, Caglar Gulcehre

EPFL · Lausanne, Switzerland · EPFL, Lausanne, Switzerland +1

Normalizing Flows Continuous Normalizing Flows Language Modeling Flow-Based Generative Modeling

Apr 27, 2026cs.LG

CoreFlow: Low-Rank Matrix Generative Models

Learning matrix-valued distributions from high-dimensional and possibly incomplete training data is challenging: ambient-space generative modeling is computationally expensive and statistically fragile when the matrix dimension is large but the sample size is limited. We propose CoreFlow, a geometry-preserving low-rank flow model that learns shared row/column subspaces across the matrix distribution, and then trains a continuous normalizing flow only on the induced low-dimensional core. CoreFlow is designed for settings where shared low-rank matrix geometry is present, especially in high-dimensional limited-sample regimes. This separates shared matrix geometry from sample-specific variation, preserves matrix structure, and substantially improves training efficiency. The same framework also handles incomplete training matrices through masked Riemannian updates and iterative completion. Across real and synthetic benchmarks, CoreFlow substantially improves spectral and moment-level generation quality in few-sample regimes while remaining competitive in data-rich settings, even under compression to 9% of the ambient dimension and with up to 40% missing training entries.

Dongze Wu, Linglingzhi Zhu, Yao Xie

H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA

Generative Modeling Continuous Normalizing Flows Flow-Based Generative Modeling Matrix Completion

Apr 19, 2026cs.LG

Prior-Fitted Functional Flow: In-Context Generative Models for Pharmacokinetics

We introduce Prior-Fitted Functional Flows, a generative foundation model for pharmacokinetics that enables zero-shot population synthesis and individual forecasting without manual parameter tuning. We learn functional vector fields, explicitly conditioned on the sparse, irregular data of an entire study population. This enables the generation of coherent virtual cohorts as well as forecasting of partially observed patient trajectories with calibrated uncertainty. We construct a new open-access literature corpus to inform our priors, and demonstrate state-of-the-art predictive accuracy on extensive real-world datasets.

César Ojeda, Niklas Hartung, Wilhelm Huisinga +6

University of Potsdam, Germany · Technical University Berlin, Germany · Freie Universität Berlin, Germany +2

Zero-Shot Time Series Forecasting Healthcare Synthetic Data Generation Continuous Normalizing Flows Prior-Data Fitted Networks Flow-Based Generative Modeling+1

Oct 3, 2024stat.ML

Local Flow Matching Generative Models

Flow Matching (FM) is a simulation-free method for learning a continuous, invertible flow that interpolates between two distributions, and in particular generates data from noise. Inspired by the variational nature of the diffusion process as a gradient flow, we introduce a stepwise FM model, Local Flow Matching (LFM), which sequentially learns a sequence of FM submodels, each matching a diffusion process up to the time-step size in the data-to-noise direction. In each step, the two distributions to be interpolated by the sub-flow model are closer than those in the full-flow matching model, which interpolates data to noise distributions, enabling smaller models with more efficient training. This variational perspective also allows us to prove a theoretical generation guarantee for the proposed flow model in terms of the

χ^2

-divergence between the generated and true data distributions, leveraging the contraction property of the diffusion process. In practice, the stepwise structure of LFM is naturally amenable to model distillation, and various distillation techniques can be applied to accelerate generation. We empirically demonstrate that LFM achieves competitive generative performance compared to FM on unconditional generation of tabular and image datasets, and on conditional generation of robotic manipulation policies.

Chen Xu, Xiuyuan Cheng, Yao Xie

H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology. · Department of Mathematics, Duke University

Flow Matching Generative Modeling Continuous Normalizing Flows Flow-Based Generative Modeling

Date pendingcs.LG

Autonomous-Flow-Based Generation

We show that using autonomous-flow-based generation, one can universally approximate orientation-preserving diffeomorphisms defined on the cube by Neural ODEs with rate

\mathcal{O}(P^{-1/d})

with

P

parameters. On the other hand, we show that by using only a single autonomous flow, the class of Neural ODEs is nowhere dense on the cube in dimension

d \ge 2

. Under a compact-support

_\mathrm{id}

condition on

(0,1)^d

, we show that using autonomous-flow-based generation, one can universally approximate compactly supported

_\mathrm{id}

diffeomorphisms on

(0,1)^d

for any dimension with rate

\mathcal{O}((\frac{P}{\log P})^{-2/d})

with

P

parameters and for compactly supported

_\mathrm{id}

homeomorphisms on

(0,1)^d

in dimension

d \geq 5

with rate

\mathcal{O}(P^{-1/(d+1)})

with

P

parameters and by a composition of at most

I_d

autonomous Neural ODEs with the same support

_\mathrm{id}

, where

I_d

depends only on the dimension. Moreover, we show that the class of single autonomous flows compactly supported

_\mathrm{id}

on

(0,1)^d

is meagre in the space of compactly supported

_\mathrm{id}

homeomorphisms on

(0,1)^d

for

d\ge 2

. By linearly lifting the domain into one higher dimension, we obtain a universal approximation result for Lipschitz functions compactly supported on

(0,1)^d

with rate

\mathcal{O}(P^{-1/(d+1)})

with

P

parameters.

Hossein Rouhvarzi, Anastasis Kratsios

McMaster University and the Vector Institute Department of Mathematics 1280 Main Street West, Hamilton, Ontario, L8S 4K1, Canada

Universal Approximation Continuous Normalizing Flows Neural Network Approximation Theory Neural ODEs